Adams-Basforth Methods

The Adams-Bashforth methods are multistep methods obtained by the following consideration:

The solution to the initial value problem

,

satisfies the equations

and

Therefore,

We can then use the Lagrange Interpolating Polynomial through the already obtained data points ( ) , ( ),..., ( ) to estimate the value of the integral. Here, we assume a fixed stepsize .

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For example, when using a quadratic interpolating polynomial through ( ) , ( ) and ( )yields:

> P[2]:=interp([t[i]-2*h,t[i]-h,t[i]],[f(t[i-2],w[i-2]),f(t[i-1],w[i-1]),f(t[i],w[i])],tt);

> int1:=int(P[2],tt=t[i]..t[i]+h);

> simplify(int1);

Which yields the Three-step Adams-Bashforth Method:

.

In addition, the local truncation error can be determined from the error term of the Lagrange Interpolating Polynomial:

> lagerr:=(D@@3)(f(xi[i],y(xi[i])))*(tt-t[i])*(tt-t[i]+h)*(tt-t[i]+2*h)/3!;

Since does not change sign over the integration interval , we can use the Weighted Mean Value Theorem so that there exists a in the interval such that

> errint:=Int(lagerr,tt=t[i]..t[i]+h):

> errint=(D@@3)(f(nu[i],y(nu[i])))/3!*Int((tt-t[i])*(tt-t[i]+h)*(tt-t[i]+2*h),tt=t[i]..t[i]+h);

This yields the error,

> err:=(D@@3)(f(nu[i],y(nu[i])))/3!*int((tt-t[i])*(tt-t[i]+h)*(tt-t[i]+2*h),tt=t[i]..t[i]+h);

The truncation error is then given by

,

with the formula on the right-hand-side of the Three-step Adams-Bashforth method, or

.

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Similarly, one can obtain formulae and expressions for the corresponding truncation errors for other Adams-Bashforth methods. Some of these will be seen in the computer session.

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